Question: James lives in San Francisco and works in Mountain View. In the morning, he has $3$ transportation options (bus, cab, or train) to work, and in the evening he has the same $3$ choices for his trip home. If James randomly chooses his ride in the morning and in the evening, what is the probability that he'll take the same mode of transportation twice?
Explanation: $\text{Probability} = \dfrac{\text{Favorable combinations}}{\text{Total possible combinations}}$ There are $3$ possible choices for each trip, so there are $3\times3=9$ total possible combinations. If James chooses randomly, all combinations are equally likely. Each path through the tree represents one possible outcome. The green paths show the $3$ favorable outcomes. $\text{B }$ $\text{C }$ $\text{T }$ $\text{Ride to work}$ $\text{B }$ $\text{C }$ $\text{T }$ $\text{B }$ $\text{C }$ $\text{T }$ $\text{B }$ $\text{C }$ $\text{T }$ $\text{Ride home}$ The probability that James will take the same mode of transportation twice is $3$ out of $9$, or $\dfrac39$. We can simplify this fraction to $\dfrac13$.